Torsion & Compression/Extension In A Spring

A conventional spiral spring with a weight will have two components, torsion and extension.
In general, if the mass has a diameter and moment of inertia, the
torsion frequency will not match with the compression - extension frequency.
The length of the spring can be changed until the frequencies match; the mass rotates up and down in the same spiral path.
Has anyone down anything analytical with this?
Bret Cahill
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wrote:

If I remember, when you match the rotation and the vertical oscillation frequecies, the spring moves between the two modes.
Brian Whatcott Altus OK
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Superglue or tape about 9 coils of a plastic Slinky jr to an empty soda can and hang it from a support, i. e., a desk lamp so you can fine tune the length of the spring.
Pull it 3" - 4" down to the desk and rotate half a turn in the direction that tightens / increases the number of coils.
Torsion and extension are linked so it's a messy dynamics problem if the frequencies are different, but it's easy to see from the jerking and rapidly changing motion.
When you get it to oscillate back and forth at the same frequency as the vertical motion and everything is smooth, the frequencies are the same and a point on the can always traces out the same spiral path.
What's the relation between torsion spring constant kt and extension spring constant k, mass m and moment of inertia mi?
omega^2 = k/m = kt/mi ?
Rotation isn't an issue with crest to crest wave springs but this must have been studied somewhere because of concern about wear on coil springs or their supports.
Bret Cahill
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I'm really struggling with this result. If you have a 2dof system and then tune it so that the frequency of each mode is identical, then the mode shapes change and the response splits into two different mode shapes, and two frequency peaks again. Check out the theory of harmonic absorbers.
Having said that, harmonic absorbers typical use the same coordinate system for both modes. You have chosen to use coordinate systems which are practically orthogonal (ie there is no cross coupling).
Ah OK I bet what you get is one mode in which the torsion and extension are in phase, and another in which the torsion and deflection are out of phase, at a different frequency.
2 dof systems are analysable by hand, even with damping. 3dof with damping gets too messy to do by hand, at least when I do it.
Cheers
Greg Locock
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I get a frequency about twice as high supporting the can on a needle bearing -- pure torsion w/o extension.
Bret Cahill
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